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ElectricalCircuitTheoryandTechnology

Afullycomprehensivetextforcoursesinelectrical principles,circuittheoryandelectricaltechnology,providing800workedexamplesandover1,350further problemsforstudentstoworkthroughattheirown pace.ThisbookisidealforstudentsstudyingengineeringforthefirsttimeaspartofBTECNational andotherpre-degreevocationalcourses,aswellas HigherNationals,FoundationDegreesandfirst-year undergraduatemodules.

JohnBird,BSc(Hons),CEng,CSci,CMath,FITE, FIMA,FCollT,istheformerHeadofAppliedElectronicsintheFacultyofTechnologyatHighburyCollege, Portsmouth,UK.Morerecentlyhehascombinedfreelancelecturingandexamining,andistheauthorof over130textbooksonengineeringandmathematicalsubjectswithworldwidesalesofoveronemillioncopies.HeiscurrentlylecturingattheDefence SchoolofMarineandAirEngineeringintheDefence CollegeofTechnicalTrainingatHMSSultan,Gosport, Hampshire,UK.

InMemoryofElizabeth

ElectricalCircuitTheoryandTechnology

Sixthedition

JohnBird

Sixtheditionpublished2017

byRoutledge

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© 2017JohnBird

TherightofJohnBirdtobeidentifiedasauthorofthisworkhasbeenassertedbyhiminaccordancewithsections77and78 oftheCopyright,DesignsandPatentsAct1988.

Allrightsreserved.Nopartofthisbookmaybereprintedorreproducedorutilizedinanyformorbyanyelectronic,mechanical, orothermeans,nowknownorhereafterinvented,includingphotocopyingandrecording,orinanyinformationstorageor retrievalsystem,withoutpermissioninwritingfromthepublishers.

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Names:Bird,J.O.,author.

Title:Electricalcircuittheoryandtechnology / JohnBird.

Description:6thed. | NewYork:Routledge,[2017] | Includesindex. Identifiers:LCCN2016038154| ISBN9781138673496 | ISBN9781315561929

Subjects:LCSH:Electriccircuits. | Electricalengineering. Classification:LCCTK454.B482017 | DDC621.319/2–dc23

LCrecordavailableathttps://lccn.loc.gov/2016038154

ISBN:978-1-138-67349-6(pbk)

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3.8Electricalpowerandenergy 54

3.9Summaryofterms,unitsandtheirsymbols55 4Anintroductiontoelectriccircuits 56

4.1Standardsymbolsforelectrical components 57

4.2Electriccurrentandquantityof electricity 57

4.3Potentialdifferenceandresistance58

4.4Basicelectricalmeasuring instruments 58

4.5Linearandnon-lineardevices 59

4.6Ohm’slaw 59

4.7Multiplesandsub-multiples 59

4.8Conductorsandinsulators 61

4.9Electricalpowerandenergy 61

4.10Maineffectsofelectriccurrent 64

4.11Fuses 64

4.12Insulationandthedangersofconstant highcurrentflow 64

5Resistancevariation 65

5.1Resistorconstruction 66

5.2Resistanceandresistivity 66

5.3Temperaturecoefficientofresistance68

5.4Resistorcolourcodingandohmicvalues70

6Batteriesandalternativesourcesofenergy73

6.1Introductiontobatteries 74

6.2Somechemicaleffectsofelectricity74

6.3Thesimplecell 75

6.4Corrosion 76

6.5E.m.f.andinternalresistanceofacell76

6.6Primarycells 78

6.7Secondarycells 79

6.8Lithium-ionbatteries 81

6.9Cellcapacity 84

6.10Safedisposalofbatteries 84

6.11Fuelcells 84

6.12Alternativeandrenewableenergysources85

6.13Solarenergy 86

7Seriesandparallelnetworks

7.1Seriescircuits

7.2Potentialdivider

7.3Parallelnetworks

7.4Currentdivision

7.5Loadingeffect

7.6Potentiometersandrheostats

7.7Relativeandabsolutevoltages 103

7.8Earthpotentialandshortcircuits104

7.9Wiringlampsinseriesandinparallel104

8Capacitorsandcapacitance

8.1Introductiontocapacitors

8.2Electrostaticfield

8.3Electricfieldstrength

8.5Capacitors

8.6Electricfluxdensity

8.8Theparallelplatecapacitor 111

8.9Capacitorsconnectedinparallel andseries 112

8.10Dielectricstrength

8.11Energystored

8.12Practicaltypesofcapacitor

8.13Supercapacitors

8.14Dischargingcapacitors

9Magneticcircuits

9.1Introductiontomagnetismand magneticcircuits

9.2Magneticfields

9.3Magneticfluxandfluxdensity

9.4Magnetomotiveforceandmagnetic fieldstrength

9.5Permeabilityand B–H curves

9.6Reluctance

9.7Compositeseriesmagneticcircuits129

9.8Comparisonbetweenelectricaland magneticquantities 132

9.9Hysteresisandhysteresisloss

10.3Forceonacurrent-carryingconductor139

10.4Principleofoperationofasimple d.c.motor 142

10.5Principleofoperationofamoving-coil instrument 143

11Electromagneticinduction 145

11.1Introductiontoelectromagneticinduction146 11.2Lawsofelectromagneticinduction147 11.3Rotationofaloopinamagneticfield150 11.4Inductance 151

11.5Inductors 152 11.6Energystored 153 11.7Inductanceofacoil 153 11.8Mutualinductance 155

12Electricalmeasuringinstrumentsand measurements 158

12.1Introduction 159 12.2Analogueinstruments 159 12.3Shuntsandmultipliers 159

12.4Electronicinstruments 161

12.5Theohmmeter 161 12.6Multimeters 162 12.7Wattmeters 162

12.8Instrument‘loading’effect 162 12.9Theoscilloscope 164

12.10Virtualtestandmeasuringinstruments169 12.11Virtualdigitalstorageoscilloscopes170 12.12Waveformharmonics 173 12.13Logarithmicratios 174

12.14Nullmethodofmeasurement 176 12.15Wheatstonebridge 177 12.16D.c.potentiometer 177 12.17A.c.bridges 178 12.18Measurementerrors 179

13Semiconductordiodes 182 13.1Typesofmaterial 183 13.2Semiconductormaterials 183 13.3Conductioninsemiconductormaterials185 13.4Thep–njunction 185 13.5Forwardandreversebias 186 13.6Semiconductordiodes 189 13.7Characteristicsandmaximumratings190 13.8Rectification 190 13.9Zenerdiodes 190 13.10Siliconcontrolledrectifiers 192 13.11Lightemittingdiodes 193 13.12Varactordiodes 193 13.13Schottkydiodes 193

14Transistors 195 14.1Transistorclassification 196 14.2Bipolarjunctiontransistors(BJTs)196 14.3Transistoraction 197 14.4Leakagecurrent 198 14.5Biasandcurrentflow 199 14.6Transistoroperatingconfigurations199

14.7Bipolartransistorcharacteristics200

14.8Transistorparameters 201

14.9Currentgain 202

14.10TypicalBJTcharacteristicsandmaximum ratings 203

14.11Fieldeffecttransistors 204

14.12Fieldeffecttransistorcharacteristics205

14.13TypicalFETcharacteristicsandmaximum ratings 206

14.14Transistoramplifiers

14.15Loadlines

17.8Q-factor

18Single-phaseparallela.c.circuits

18.2 R –L parallela.c.circuit

18.3 R –C parallela.c.circuit

19D.c.transients

Part3Electricalprinciplesand

15D.c.circuittheory

15.1Introduction 219

15.2Kirchhoff’slaws

15.3Thesuperpositiontheorem 224

15.4Generald.c.circuittheory 226

15.5Thévenin’stheorem 228

15.6Constant-currentsource 233

15.7Norton’stheorem 233

15.8ThéveninandNortonequivalentnetworks236

15.9Maximumpowertransfertheorem239

16Alternatingvoltagesandcurrents

16.1Introduction

16.2Thea.c.generator

16.4A.c.values 245

16.5Electricalsafety–insulationandfuses248

16.6Theequationofasinusoidalwaveform248

16.7Combinationofwaveforms 251

16.8Rectification 254

16.9Smoothingoftherectifiedoutputwaveform255

17Single-phaseseriesa.c.circuits

17.1Purelyresistivea.c.circuit

17.2Purelyinductivea.c.circuit

17.3Purelycapacitivea.c.circuit

17.4 R –L seriesa.c.circuit

17.5

17.6

17.7Seriesresonance

19.4Transientcurvesfora C –R circuit296 19.5Dischargingacapacitor

19.7Currentgrowthinan L

R circuit302

19.8Timeconstantforan L–R circuit303 19.9Transientcurvesforan L–R circuit303 19.10Currentdecayinan L–R circuit 305 19.11Switchinginductivecircuits 307 19.12Theeffectoftimeconstantona rectangularwaveform 307

20Operationalamplifiers

309

20.1Introductiontooperationalamplifiers310 20.2Someopampparameters 311 20.3Opampinvertingamplifier 312 20.4Opampnon-invertingamplifier 314 20.5Opampvoltage-follower 315 20.6Opampsummingamplifier 315 20.7Opampvoltagecomparator 316 20.8Opampintegrator 317 20.9Opampdifferentialamplifier 318 20.10Digitaltoanalogue(D/A)conversion320 20.11Analoguetodigital(A/D)conversion320

21Waysofgeneratingelectricity–thepresent andthefuture 323

21.1Introduction 324

21.2Generatingelectricalpowerusingcoal324

21.3Generatingelectricalpowerusingoil326

21.4Generatingelectricalpowerusing naturalgas 327

21.5Generatingelectricalpowerusingnuclear energy 328

21.6Generatingelectricalpowerusinghydro power 329

21.7Generatingelectricalpowerusingpumped storage 330

21.8Generatingelectricalpowerusingwind331

21.9Generatingelectricalpowerusingtidal power 331

21.10Generatingelectricalpowerusingbiomass333

21.11Generatingelectricalpowerusingsolar energy 333

21.12Harnessingthepowerofwind,tideand sunonan‘energyisland’–afuture possibility? 334

22Three-phasesystems 336

22.1Introduction 337

22.2Three-phasesupply 337

22.3Starconnection 337

22.4Deltaconnection 340

22.5Powerinthree-phasesystems 342

22.6Measurementofpowerinthree-phase systems 343

22.7Comparisonofstaranddeltaconnections348

22.8Advantagesofthree-phasesystems348

23Transformers 349

23.1Introduction 350

23.2Transformerprincipleofoperation350

23.3Transformerno-loadphasordiagram352

23.4E.m.f.equationofatransformer 354

23.5Transformeron-loadphasordiagram356

23.6Transformerconstruction 357

23.7Equivalentcircuitofatransformer358

23.8Regulationofatransformer 359

23.9Transformerlossesandefficiency360

23.10Resistancematching 363

23.11Autotransformers 365

23.12Isolatingtransformers 367

23.13Three-phasetransformers 367

23.14Currenttransformers 368

23.15Voltagetransformers 369

RevisionTest6 370

24D.c.machines 371

24.1Introduction 372

24.2Theactionofacommutator 372

24.3D.c.machineconstruction 373

24.4Shunt,seriesandcompoundwindings373

24.5E.m.f.generatedinanarmaturewinding374

24.6D.c.generators 375

24.7Typesofd.c.generatorandtheir characteristics 376

24.8D.c.machinelosses 380

24.9Efficiencyofad.c.generator 380

24.10D.c.motors 381

24.11Torqueofad.c.machine 382

24.12Typesofd.c.motorandtheir characteristics 383

24.13Theefficiencyofad.c.motor 387

24.14D.c.motorstarter 389

24.15Speedcontrolofd.c.motors 390

24.16Motorcooling 392

25Three-phaseinductionmotors 393

25.1Introduction 394

25.2Productionofarotatingmagneticfield394

25.3Synchronousspeed 396

25.4Constructionofathree-phaseinduction motor 397

25.5Principleofoperationofathree-phase inductionmotor 397

25.6Slip 398

25.7Rotore.m.f.andfrequency 399

25.8Rotorimpedanceandcurrent 400

25.9Rotorcopperloss 400

25.10Inductionmotorlossesandefficiency401

25.11Torqueequationforaninductionmotor402

25.12Inductionmotortorque–speed characteristics 404

25.13Startingmethodsforinductionmotors405

25.14Advantagesofsquirrel-cageinduction motors 406

25.15Advantagesofwoundrotorinduction motor 407

25.16Doublecageinductionmotor 407

25.17Usesofthree-phaseinductionmotors407

26.5Multiplicationanddivisionusingcomplex numbersinpolarform 419

26.6DeMoivre’stheorem–powersandroots ofcomplexnumbers 420

27Applicationofcomplexnumberstoseries a.c.circuits 423

27.1Introduction 423

27.2Seriesa.c.circuits 424

27.3Furtherworkedproblemsonseries a.c.circuits 430

28Applicationofcomplexnumberstoparallel a.c.networks 435

28.1Introduction 435

28.2Admittance,conductanceandsusceptance436

28.3Parallela.c.networks 439

28.4Furtherworkedproblemsonparallel a.c.networks 443

29Powerina.c.circuits 446

29.1Introduction 446

29.2Determinationofpowerina.c.circuits447

29.3Powertriangleandpowerfactor 449

29.4Useofcomplexnumbersfor determinationofpower 450

29.5Powerfactorimprovement 454

RevisionTest8 459

30A.c.bridges 460

30.1Introduction 461

30.2Balanceconditionsforana.c.bridge461

30.3Typesofa.c.bridgecircuit 462

30.4Workedproblemsona.c.bridges467

31SeriesresonanceandQ-factor 471

31.1Introduction 472

31.2Seriesresonance 472

31.3Q-factor 474

31.4Voltagemagnification 476

31.5Q-factorsinseries 478

31.6Bandwidth 479

31.7Smalldeviationsfromtheresonant frequency 483

32ParallelresonanceandQ-factor 486

32.1Introduction 486

32.2The LR –C parallelnetwork 487

32.3Dynamicresistance 488

32.4The LR –CR parallelnetwork 488

32.5Q-factorinaparallelnetwork 489

32.6Furtherworkedproblemsonparallel resonanceandQ-factor 493

33Introductiontonetworkanalysis 497

33.1Introduction 497

33.2Solutionofsimultaneousequationsusing determinants 498

33.3NetworkanalysisusingKirchhoff’slaws499

34Mesh-currentandnodalanalysis 507

34.1Mesh-currentanalysis 507

34.2Nodalanalysis 511

35Thesuperpositiontheorem 518

35.1Introduction 518

35.2Usingthesuperpositiontheorem518

35.3Furtherworkedproblemsonthe superpositiontheorem 523

36Thévenin’sandNorton’stheorems 528

36.1Introduction 528

36.2Thévenin’stheorem 529

36.3FurtherworkedproblemsonThévenin’s theorem 535

36.4Norton’stheorem 539

36.5ThéveninandNortonequivalentnetworks546

RevisionTest10 551

37Delta–starandstar–deltatransformations552

37.1Introduction 552

37.2Deltaandstarconnections 552

37.3Delta–startransformation 553

37.4Star–deltatransformation 561

38Maximumpowertransfertheoremsand impedancematching 565

38.1Maximumpowertransfertheorems566

38.2Impedancematching 571

RevisionTest11 574

39Complexwaveforms 575

39.1Introduction 576

39.2Thegeneralequationforacomplex waveform 576

39.3Harmonicsynthesis 577

39.4Fourierseriesofperiodicandnon-periodic functions 585

39.5EvenandoddfunctionsandFourierseries overanyrange 590

39.6R.m.s.value,meanvalueandtheform factorofacomplexwave 594

39.7Powerassociatedwithcomplexwaves597

39.8Harmonicsinsingle-phasecircuits599

39.9Furtherworkedproblemsonharmonics insingle-phasecircuits 602

39.10Resonanceduetoharmonics 606

39.11Sourcesofharmonics 608

40Anumericalmethodofharmonicanalysis612

40.1Introduction 612

40.2Harmonicanalysisondatagivenintabular orgraphicalform 612

40.3Complexwaveformconsiderations616

41Magneticmaterials 619

41.1Revisionoftermsandunitsusedwith magneticcircuits 620

41.2Magneticpropertiesofmaterials621

41.3Hysteresisandhysteresisloss 622

41.4Eddycurrentloss 626

41.5Separationofhysteresisandeddycurrent losses 629

41.6Non-permanentmagneticmaterials631

41.7Permanentmagneticmaterials 633

RevisionTest12 634

42Dielectricsanddielectricloss 635

42.1Electricfields,capacitanceandpermittivity635

42.2Polarization 636

42.3Dielectricstrength 636

42.4Thermaleffects 637

42.5Mechanicalproperties 638

42.6Typesofpracticalcapacitor 638

42.7Liquiddielectricsandgasinsulation638

42.8Dielectriclossandlossangle 638

43Fieldtheory 642

43.1Fieldplottingbycurvilinearsquares643

43.2Capacitancebetweenconcentriccylinders646

43.3Capacitanceofanisolatedtwinline651

43.4Energystoredinanelectricfield654

43.5Inducede.m.f.andinductance 656

43.6Inductanceofaconcentriccylinder(or coaxialcable) 656

43.7Inductanceofanisolatedtwinline659

43.8Energystoredinanelectromagneticfield662

44Attenuators 665

44.1Introduction 666

44.2Characteristicimpedance 666

44.3Logarithmicratios 668

44.4SymmetricalT-and π -attenuators670

44.5Insertionloss 675

44.6AsymmetricalT-and π -sections678

44.7TheL-sectionattenuator 681

44.8Two-portnetworksincascade 683

44.9 ABCD parameters 686

44.10 ABCD parametersfornetworks 689

44.11Characteristicimpedanceintermsof ABCD parameters 695 RevisionTest13 697

45Filternetworks 698

45.1Introduction 698

45.2Basictypesoffiltersections 699

45.3Thecharacteristicimpedanceandthe attenuationoffiltersections 701

45.4Laddernetworks 702

45.5Low-passfiltersections 703

45.6High-passfiltersections 709

45.7Propagationcoefficientandtimedelayin filtersections 714

45.8‘m-derived’filtersections 720

45.9Practicalcompositefilters 725

46Magneticallycoupledcircuits 728

46.1Introduction 728

46.2Self-inductance 728

46.3Mutualinductance 729

46.4Couplingcoefficient 730

46.5Coilsconnectedinseries 731

46.6Coupledcircuits 734

46.7Dotruleforcoupledcircuits 739

47Transmissionlines 746

47.1Introduction 746

47.2Transmissionlineprimaryconstants747

47.3Phasedelay,wavelengthandvelocityof propagation 748

47.4Currentandvoltagerelationships749

47.5Characteristicimpedanceand propagationcoefficientintermsofthe primaryconstants 751

47.6Distortionontransmissionlines 755

47.7Wavereflectionandthereflection coefficient 757

47.8Standing-wavesandthestanding-wave ratio 760

48TransientsandLaplacetransforms 765

48.1Introduction 766

48.2Responseof R –C seriescircuittoastep input 766

48.3Responseof R –L seriescircuittoastep input 768

48.4 L–R –C seriescircuitresponse 771

48.5IntroductiontoLaplacetransforms774

48.6InverseLaplacetransformsandthe solutionofdifferentialequations779

48.7Laplacetransformanalysisdirectlyfrom thecircuitdiagram 784

Preface

ElectricalCircuitTheoryandTechnology6thEdition providescoverageforawiderangeofcoursesthatcontainelectricalprinciples,circuittheoryandtechnology intheirsyllabuses,from introductorytodegreelevel –andincludingEdexcelBTECLevels2to5NationalCertificate/Diploma,HigherNationalCertificate/Diploma andFoundationdegreeinEngineering

Inthisnewsixthedition, newmaterialadded includes somemathematicsrevisionneededforelectricaland electronicprinciples,waysofgeneratingelectricity–thepresentandthefuture(includingmoreonrenewableenergy),moreonlithium-ionbatteries,alongwith otherminormodifications.

Thetextissetoutin fiveparts asfollows:

PART1,comprisingchapters1to12,involves Revision ofsomeBasicMathematics neededforElectricaland ElectronicPrinciples.

PART2,involvingchapters3to14,contains Basic ElectricalEngineeringPrinciples whichanystudent wishingtoprogressinelectricalengineeringwouldneed toknow.Anintroductiontounits,electricalcircuits, resistancevariation,batteriesandalternativesources ofenergy,seriesandparallelcircuits,capacitorsand capacitance,magneticcircuits,electromagnetism,electromagneticinduction,electricalmeasuringinstruments andmeasurements,semiconductordiodesandtransistorsareallincludedinthissection.

PART3,involvingchapters15to25,contains ElectricalPrinciplesandTechnology suitableforNational Certificate,NationalDiplomaandCityandGuilds coursesinelectricalandelectronicengineering.D.c. circuittheory,alternatingvoltagesandcurrents,singlephaseseriesandparallelcircuits,d.c.transients, operationalamplifiers,waysofgeneratingelectricity, three-phasesystems,transformers,d.c.machinesand three-phaseinductionmotorsareallincludedinthis section.

PART4,involvingchapters26to48,contains AdvancedCircuitTheoryandTechnology suitable forDegree,Foundationdegree,HigherNationalCertificate/DiplomaandCityandGuildscoursesinelectrical

andelectronic/telecommunicationsengineering.The threeearliersectionsofthebookwillprovideavaluable reference/revisionforstudentsatthislevel.

Complexnumbersandtheirapplicationtoseriesand parallelnetworks,powerina.c.circuits,a.c.bridges, seriesandparallelresonanceandQ-factor,network analysisinvolvingKirchhoff’slaws,meshandnodal analysis,thesuperpositiontheorem,Thévenin’sand Norton’stheorems,delta-starandstar-deltatransforms, maximumpowertransfertheoremsandimpedance matching,complexwaveforms,Fourierseries,harmonicanalysis,magneticmaterials,dielectricsand dielectricloss,fieldtheory,attenuators,filternetworks, magneticallycoupledcircuits,transmissionlinetheory andtransientsandLaplacetransformsareallincluded inthissection.

PART5 providesashort GeneralReference forstandardelectricalquantities–theirsymbolsandunits,the Greekalphabet,commonprefixesandresistorcolour codingandohmicvalues.

Atthebeginningofeachofthe48chaptersabrief explanationastowhyitisimportanttounderstand thematerialcontainedwithinthatchapterisincluded, togetherwithalistof learningobjectives

Attheendofeachofthefirstfourpartsofthetextisa handyreferenceofthe mainformulae used. ThereareanumberofInternetdownloadsfreelyavailabletobothstudentsandlecturers/instructors;theseare listedonpagexiii.

Itisnotpossibletoacquireathoroughunderstanding ofelectricalprinciples,circuittheoryandtechnology withoutworkingthroughalargenumberofnumerical problems.Itisforthisreasonthat ElectricalCircuit TheoryandTechnology6thEdition containsnearly 800 detailedworkedproblems,togetherwithsome 1350 furtherproblems(withanswersatthebackofthe book),arrangedwithin 202PracticeExercises that appeareveryfewpagesthroughoutthetext.Some 1153 linediagrams furtherenhancetheunderstandingofthe theory.

FourteenRevisionTests havebeenincluded,interspersedwithinthetexteveryfewchapters.Forexample, RevisionTest1testsunderstandingofchapters3to 6,RevisionTest2testsunderstandingofchapters7 to9,RevisionTest3testsunderstandingofchapters 10to14,andsoon.TheseRevisionTestsdonothave answersgivensinceitisenvisagedthatlecturers/instructorscouldsettheRevisionTestsforstudentstoattempt aspartoftheircoursestructure.Lecturers/instructors mayobtainacomplimentarysetofsolutionsoftheRevisionTestsinan Instructor’sManual availablefromthe publishersviatheinternet–seebelow.

Learningbyexample isattheheartof Electrical CircuitTheoryandTechnology6thEdition

JOHNBIRD

RoyalNavalDefenceCollegeofMarineandAir Engineering,HMSSultan, formerlyUniversityofPortsmouth andHighburyCollege,Portsmouth

JohnBirdistheformerHeadofAppliedElectronics intheFacultyofTechnologyatHighburyCollege, Portsmouth,UK.Morerecently,hehascombined freelancelecturingattheUniversityofPortsmouth withExaminerresponsibilitiesforAdvancedMathematicswithCityandGuilds,andexaminingfor theInternationalBaccalaureate.Heistheauthor ofsome130textbooksonengineeringandmathematicalsubjectswithworldwidesalesofoverone millioncopies.Heiscurrentlylecturingatthe DefenceSchoolofMarineandAirEngineeringin theDefenceCollegeofTechnicalTrainingatHMS Sultan,Gosport,Hampshire,UK.

FreeWebdownloads

Thefollowingsupportmaterialisavailablefrom www.routledge.com/cw/bird

ForStudents:

1.Fullsolutionstoall1350furtherquestions inthePracticeExercises

2.Asetofformulaeforeachofthefirstfour sectionsofthetext

3.Multiplechoicequestions

4.Informationon38Engineers/Scientists mentionedinthetext

ForLecturers/Instructors:

1–4.Asperstudents1–4above

5.Fullsolutionsandmarkingschemeforeach ofthe14RevisionTests;also,eachtestmay bedownloaded.

6.LessonPlansandrevisionmaterial.Typical30-weeklessonplansfor‘Electricaland ElectronicPrinciples’,Unit6,and‘Further ElectricalPrinciples’,Unit64,areincluded, togetherwithtwopracticeexaminationquestionpapers(withsolutions)foreachofthe modules.

7.TenpracticalLaboratoryExperiments are available.Itmaybethattutorswillwant toedittheseexperimentstosuittheirown equipment/componentavailability.

8.All1153illustrationsusedinthetext maybedownloadedforuseinPowerPoint Presentations.

Chapter1

Somemathematicsrevision

Whyitisimportanttounderstand: Somemathematicsrevision

Mathematicsisavitaltoolforprofessionalandcharteredengineers.Itisusedinelectricaland electronicengineering,inmechanicalandmanufacturingengineering,incivilandstructuralengineering,innavalarchitectureandmarineengineeringandinaeronauticalandrocketengineering.In thesevariousbranchesofengineering,itisveryoftenmuchcheaperandsafertodesignyourartefactwiththeaidofmathematics–ratherthanthroughguesswork.‘Guesswork’maybereasonably satisfactoryifyouaredesigninganexactlysimilarartefactasonethathasalreadyprovensatisfactory;however,theclassificationsocietieswillusuallyrequireyoutoprovidethecalculationsproving thattheartefactissafeandsound.Moreover,thesecalculationsmaynotbereadilyavailabletoyou andyoumayhavetoprovidefreshcalculations,toprovethatyourartefact is‘roadworthy’.For example,ifyoudesignatallbuildingoralongbridgeby‘guesswork’,andthebuildingorbridge donotprovetobestructurallyreliable,itcouldcostyouafortunetorectifythedeficiencies.This costmaydwarftheinitialestimateyoumadetoconstructthesestructures,andcauseyoutogo bankrupt.Thus,withoutmathematics,theprospectiveprofessionalorcharteredengineerisveryseverely disadvantaged.

Knowledgeofmathematicsprovidesthebasisforallengineering.

Attheendofthischapteryoushouldbeableto:

• useacalculatorandevaluateformulae

• manipulatefractions

• understandandperformcalculationswithpercentages

• appreciateratiosanddirectandinverseproportion

• understandandusethelawsofindices

• expandequationscontainingbrackets

• solvesimpleequations

• transposeformulae

• solvesimultaneousequationsintwounknowns

ElectricalCircuitTheoryandTechnology.978-1-138-67349-6,©2017JohnBird.PublishedbyTaylor&Francis.Allrightsreserved.

1.1Useofcalculatorandevaluating formulae

Inengineering,calculationsoftenneedtobeperformed. Forsimplenumbersitisusefultobeabletousementalarithmetic.However,whennumbersarelargeran electroniccalculatorneedstobeused.

Inengineeringcalculationsitisessentialtohavea scientificnotationcalculator whichwillhaveallthe necessaryfunctionsneeded,andmore.Thischapter assumesyouhavea CASIOfx-991ESPLUScalculator,orsimilar.Ifyoucanaccuratelyuseacalculator,yourconfidencewithengineeringcalculationswill improve.

Checkthatyoucanuseacalculatorinthefollowing PracticeExercise.

PracticeExercise1Useofcalculator (Answersonpage815)

1.Evaluate 378 37 298 651 + 45 64 94 562

2.Evaluate 17 35 × 34 27 41.53 ÷ 3.76 correctto3decimal places

3.Evaluate (4 527 + 3 63) (452 51 ÷ 34 75) + 0 468correct to5significantfigures

4.Evaluate52 34 (912 5 ÷ 41 46) (24 6 13 652) correctto 3decimalplaces

5.Evaluate 52.14 × 0.347 × 11.23 19.73 ÷ 3.54 correctto4 significantfigures

6.Evaluate6.85 2 correctto3decimalplaces

7.Evaluate (0.036 )2 inengineeringform

8.Evaluate1 3 3

9.Evaluate (0 38)3 correctto4decimalplaces

10.Evaluate (0 018 )3 inengineeringform

11.Evaluate 1 0.00725 correctto1decimalplace

12.Evaluate 1 0 065 1 2 341 correctto4significantfigures

13.Evaluate2 1 4

14.Evaluate (0 22)5 correctto5significant figuresinengineeringform

15.Evaluate (1.012)7 correctto4decimalplaces

16.Evaluate1.1 3 + 2.94 4.42 correctto4significantfigures

17.Evaluate √34528correctto2decimalplaces

18.Evaluate 3 √17correctto3decimalplaces

19.Evaluate 6 √2451 4 √46correctto3decimal places

Expresstheanswerstoquestions20to23in engineeringform.

20.Evaluate5 × 10 3 × 7 × 108

21.Evaluate 6 × 103 × 14 × 10 4 2 × 106

22.Evaluate 56 43 × 10 3 × 3 × 104 8.349 × 10 3 correctto 3decimalplaces

23.Evaluate 99 × 10 5 × 6 7 × 10 3 36.2 × 10 4 correctto4 significantfigures

24.Evaluate 4 5 1 3 asadecimal,correctto4 decimalplaces

25.Evaluate 2 3 1 6 + 3 7 asafraction

26.Evaluate2 5 6 + 1 5 8 asadecimal,correctto4 significantfigures

27.Evaluate5 6 7 3 1 8 asadecimal,correctto4 significantfigures

28.Evaluate 3 4 × 4 5 2 3 ÷ 4 9 asafraction

29.Evaluate8 8 9 ÷ 2 2 3 asamixednumber

30.Evaluate3 1 5 × 1 1 3 1 7 10 asadecimal, correctto3decimalplaces

31.Evaluate 4 1 5 1 2 3 3 1 4 × 2 3 5 2 9 asadecimal, correctto3significantfigures

Inquestions32to38,evaluatecorrectto4decimal places.

32.Evaluatesin67 ◦

33.Evaluatetan71 ◦

34.Evaluatecos63 74 ◦

35.Evaluatetan39 55 ◦ sin52 53 ◦

36.Evaluatesin (0 437rad)

37.Evaluatetan (5.673rad)

38.Evaluate (sin42 6 ◦ )(tan83 2 ◦ ) cos13 8 ◦

Inquestions39to45,evaluatecorrectto4significantfigures.

39.1.59π

40.2 7(π 1)

41. π 2 √13 1

42.8.5e 2 5 43.3e (2π 1) 44. 5.52π 2e 2 × √26 73 45. ⎡ ⎣ e 2 √3 π × √8 57 ⎤ ⎦

Evaluationofformulae

Thestatement y = mx + c iscalleda formula foryin termsofm,xandc. y,m,xandcarecalled symbols Whengivenvaluesofm,xandcwecanevaluatey. Therearealargenumberofformulaeusedinengineeringandinthissectionwewillinsertnumbersinplace ofsymbolstoevaluateengineeringquantities. Herearesomepracticalexamples.Checkwithyour calculatorthatyouagreewiththeworkingandanswers.

Problem1. InanelectricalcircuitthevoltageV isgivenbyOhm’slaw,i.e.V = IR.Find,correctto 4significantfigures,thevoltagewhenI = 5 36A andR = 14 76

V = IR = I × R = 5 36 × 14 76

Hence, voltageV = 79.11V,correctto4significant figures

Problem2. Velocityvisgivenbyv = u + at.If u = 9.54m/s,a = 3.67m/s 2 andt = 7.82s,findv, correctto3significantfigures.

v = u + at = 9 54 + 3 67 × 7 82 = 9.54 + 28.6994 = 38.2394

Hence, velocityv = 38.2m/s,correctto3significant figures

Problem3. Thearea,A,ofacircleisgivenby A = π r2 .Determinetheareacorrectto2decimal places,givenradiusr = 5.23m.

A = π r 2 = π(5 23) 2 = π(27 3529)

Hence, area,A = 85.93m 2,correctto2decimalplaces

Problem4. Density = mass volume .Findthedensity whenthemassis6.45kgandthevolumeis 300 × 10 6 m 3

Density = mass volume = 6.45kg 300 × 10 6 m 3 =21500kg /m3

Problem5. Thepower,Pwatts,dissipatedinan electricalcircuitisgivenbytheformulaP = V2 R . Evaluatethepower,correctto4significantfigures, giventhatV = 230VandR = 35 63

P = V2 R = (230) 2 35.63 = 52900 35.63 = 1484.70390 ...

PressENGand1 48470390 × 10 3 appearsonthe screen

Hence, power,P = 1485Wor1.485kWcorrectto4 significantfigures.

ElectricalCircuitTheoryandTechnology

Problem6. Resistance,R ,varieswith temperatureaccordingtotheformula

R = R0 (1 + α t ).EvaluateR,correctto3significant figures,givenR 0 = 14 59, α = 0 0043andt = 80

R = R0 (1 + α t ) = 14.59[1 + (0.0043)(80)] = 14 59(1 + 0 344) = 14 59(1 344)

Hence, resistance,R = 19.6 ,correctto3significant figures

Problem7. Thecurrent,Iamperes,inana.c. circuitisgivenby:I = V (R 2 + X 2 ) Evaluatethe current,correctto2decimalplaces,when V = 250V,R = 25.0 andX = 18.0

I = V (R2 +X2 ) = 250 25.02 +18.02 =8 11534341

Hence, current,I = 8.12A,correctto2decimal places

NowtrythefollowingPracticeExercise

PracticeExercise2Evaluationofformulae (Answersonpage815)

1.TheareaAofarectangleisgivenbythe formulaA = l × b.Evaluatethearea,correct to2decimalplaces,whenl = 12.4cmand b = 5.37cm

2.ThecircumferenceCofacircleisgivenby theformulaC = 2π r.Determinethecircumference,correctto2decimalplaces,given r = 8 40mm

3.Aformulausedinconnectionwithgasesis R = PV T .EvaluateRwhenP = 1500,V = 5 andT = 200

4.Thevelocityofabodyisgivenbyv = u + at. Theinitialvelocityuismeasuredwhentime tis15secondsandfoundtobe12m/s.Ifthe accelerationais9.81m/s 2 calculatethefinal velocityv

5.CalculatethecurrentIinanelectricalcircuit, correctto3significantfigures,whenI = V/R ampereswhenthevoltageVismeasuredand

foundtobe7.2VandtheresistanceRis 17.7

6.Findthedistances,giventhats = 1 2 gt2 .Time t = 0.032secondsandaccelerationdueto gravityg = 9 81m/s 2 .Givetheanswerin millimetrescorrectto3significantfigures.

7.Theenergystoredinacapacitorisgiven byE = 1 2 CV2 joules.Determinetheenergy whencapacitanceC = 5 × 10 6 faradsand voltageV = 240V

8.FindtheareaAofatriangle,correctto1decimalplace,givenA = 1 2 bh,whenthebase lengthbis23.42mandtheheighthis53.7m

9.ResistanceR2 isgivenbyR 2 = R1 (1 + α t ) FindR2 ,correctto4significantfigures,when R1 = 220, α = 0.00027andt = 75.6

10.Density = mass volume .Findthedensity,correct to4significantfigures,whenthemass is2.462kgandthevolumeis173cm 3 .Give theanswerinunitsofkg/m 3 .Notethat 1cm3 = 10 6 m 3

11.EvaluateresistanceR T ,correctto4significantfigures,given 1 RT = 1 R1 + 1 R2 + 1 R3 whenR1 = 5 5 ,R2 = 7 42 and R3 = 12 6

12.Thepotentialdifference,Vvolts,available atbatteryterminalsisgivenbyV = E Ir. EvaluateVwhenE = 5.62,I = 0.70and R = 4.30

13.ThecurrentIamperesflowinginanumber ofcellsisgivenbyI = nE R + nr .Evaluatethe current,correctto3significantfigures,when n = 36.E = 2.20,R = 2.80andr = 0.50

14.Energy,Ejoules,isgivenbytheformula E = 1 2 LI2 .Evaluatetheenergywhen L = 5 5HandI = 1 2A

15.ThecurrentIamperesinana.c.circuit isgivenbyI = V (R2 + X2 ) .Evaluatethe

current,correctto4significantfigures,when V = 250V,R = 11.0 andX = 16.2

1.2Fractions

Anexampleofafractionis 2 3 wherethetopline,i.e.the 2,isreferredtoasthe numerator andthebottomline, i.e.the3,isreferredtoasthe denominator

A properfraction isonewherethenumeratoris smallerthanthedenominator,examplesbeing 2 3 , 1 2 , 3 8 , 5 16 ,andsoon.

An improperfraction isonewherethedenominator issmallerthanthenumerator,examplesbeing 3 2 , 2 1 , 8 3 , 16 5 ,andsoon.

Additionoffractionsisdemonstratedinthefollowing workedproblems.

Problem8. EvaluateA,givenA = 1 2 + 1 3

Thelowestcommondenominatorofthetwodenominators2and3is6,i.e.6isthelowestnumberthatboth2 and3willdivideinto.

Then 1 2 = 3 6 and 1 3 = 2 6 i.e.both 1 2 and 1 3 havethe commondenominator,namely6.

Thetwofractionscanthereforebeaddedas:

Problem9. EvaluateA,givenA = 2 3 + 3 4

Acommondenominatorcanbeobtainedbymultiplyingthetwodenominatorstogether,i.e.thecommon denominatoris3 × 4 = 12

Thetwofractionscannowbemadeequivalent,i.e. 2 3 = 8 12 and 3 4 = 9 12 sothattheycanbeeasilyaddedtogether,asfollows:

Problem10. EvaluateA,givenA = 1 6 + 2 7 + 3 2

Asuitablecommondenominatorcanbeobtainedby multiplying6 × 7 = 42,andallthreedenominators divideexactlyinto42.

Thus, 1 6 = 7 42 , 2 7 = 12 42 and 3 2 = 63 42

Hence,A = 1 6 + 2 7 + 3 2 =

Problem11. DetermineAasasinglefraction, givenA = 1 x + 2 y

Acommondenominatorcanbeobtainedbymultiplying thetwodenominatorstogether,i.e.xy

Thus, 1 x = y xy and 2

= 2x xy

Notethataddition,subtraction,multiplicationanddivisionoffractionsmaybedeterminedusinga calculator (forexample,theCASIOfx-991ESPLUS).

Locatethe and functionsonyourcalculator(thelatterfunctionisashiftfunctionfoundabove the function)andthencheckthefollowingworked problems.

Problem12. Evaluate 1 4 + 2 3 usingacalculator

(i)Press function

(ii)Typein1

(iii)Press ↓ onthecursorkeyandtypein4

(iv) 1 4 appearsonthescreen

(v)Press → onthecursorkeyandtypein+

(vi)Press function

8 ElectricalCircuitTheoryandTechnology

(vii)Typein2

(viii)Press ↓ onthecursorkeyandtypein3

(ix)Press → onthecursorkey

(x)Press = andtheanswer 11 12 appears

(xi)PressS ⇔ Dfunctionandthefractionchanges toadecimal0 9166666

Thus, 1 4 + 2 3 = 11 12 = 0.9167 asadecimal,correctto4 decimalplaces.

Itisalsopossibletodealwith mixednumbers onthe calculator.

PressShiftthenthe functionand appears.

Problem13. Evaluate5 1 5 3 3 4 usingacalculator

(i)PressShiftthenthe functionand appears onthescreen

(ii)Typein5then → onthecursorkey

(iii)Typein1and↓onthecursorkey

(iv)Typein5and5 1 5 appearsonthescreen

(v)Press → onthecursorkey

(vi)Typein–andthenpressShiftthenthe function and5 1 5 appearsonthescreen

(vii)Typein3then → onthecursorkey

(viii)Typein3and ↓ onthecursorkey

(ix)Typein4and5 1 5 3 3 4 appearsonthescreen

(x)Press = andtheanswer 29 20 appears

(xi)Pressshiftandthen S ⇔ D functionand1 9 20 appears

(xii)PressS ⇔ Dfunctionandthefractionchangesto adecimal1.45

Thus, 5 1 5 3 3 4 = 29 20 = 1 9 20 = 1 45 asadecimal

NowtrythefollowingPracticeExercise

PracticeExercise3Fractions(Answerson page815)

Inproblems1to3,evaluatethegivenfractions

1. 1 3 + 1 4

2. 1 5 + 1 4

3. 1 6 + 1 2 1 5

Inproblems4and5,useacalculatortoevaluate thegivenexpressions

4. 1 3 3 4 × 8 21 5. 3 4 × 4 5 2 3 ÷ 4 9

6.Evaluate 3 8 + 5 6 1 2 asadecimal,correctto4 decimalplaces.

7.Evaluate8 8 9 ÷ 2 2 3 asamixednumber.

8.Evaluate3 1 5 × 1 1 3 1 7 10 asadecimal,correct to3decimalplaces.

9.Determine 2 x + 3 y asasinglefraction.

1.3Percentages

Percentages areusedtogiveacommonstandard.The useofpercentagesisverycommoninmanyaspects ofcommerciallife,aswellasinengineering.Interest rates,salereductions,payrises,examsandVATareall exampleswherepercentagesareused.

Percentagesarefractionshaving100astheirdenominator.

Forexample,thefraction 40 100 iswrittenas40%and isreadas‘fortypercent’.

Theeasiestwaytounderstandpercentagesistogo throughsomeworkedexamples.

Problem14. Express0.275asapercentage 0 275 = 0 275 × 100% = 27 5%

Problem15. Express17.5%asadecimalnumber

17.5% = 17.5

100 = 0.175

Problem16. Express 5 8 asapercentage

5 8 = 5 8 × 100% = 500 8 % = 62 .5%

Problem17. Intwosuccessivetestsastudent gainsmarksof57/79and49/67.Isthesecondmark betterorworsethanthefirst?

57/79 = 57 79 = 57 79 × 100% = 5700 79 %

= 72 15%correctto2decimalplaces

49/67 = 49 67 = 49 67 × 100% = 4900 67 %

= 73.13%correctto2decimalplaces

Hence, thesecondtestmarkismarginallybetterthan thefirsttest.

Thisquestiondemonstrateshowmucheasieritis tocomparetwofractionswhentheyareexpressedas percentages.

Problem18. Express75%asafraction

75% = 75 100 = 3 4

Thefraction 75 100 isreducedtoitssimplestformby cancelling,i.e.dividingnumeratoranddenominatorby 25.

Problem19. Express37.5%asafraction

37.5% = 37 5 100

= 375 1000 bymultiplyingnumerator anddenominatorby10

= 15 40 bydividingnumerator anddenominatorby25

= 3 8 bydividingnumerator anddenominatorby5

Problem20. Find27%of£65

27%of£65 = 27 100 × 65 = £17 55 bycalculator

Problem21. A160GBiPodisadvertisedas costing£190excludingVAT.IfVATisaddedat 20%,whatwillbethetotalcostoftheiPod?

VAT = 20%of£190 = 20 100 × 190 = £38

TotalcostofiPod = £190 + £38 = £228

Aquickermethodtodeterminethetotalcostis: 1.20 × £190 = £228

Problem22. Express23cmasapercentageof 72cm,correcttothenearest1%

23cmasapercentageof72cm

= 23 72 × 100% = 31.94444 ...... %

= 32%correcttothenearest1%

Problem23. Aboxofscrewsincreasesinprice from£45to£52.Calculatethepercentagechange incost,correctto3significantfigures.

%change = newvalue–originalvalue originalvalue × 100%

= 52 45 45 × 100% = 7 45 × 100 = 15.6%

= percentagechangeincost

Problem24. Adrillingspeedshouldbesetto400 rev/min.Thenearestspeedavailableonthemachine is412rev/min.Calculatethepercentageover-speed.

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ElectricalCircuitTheoryandTechnology

%over-speed = availablespeed–correctspeed correctspeed × 100% = 412

× 100% = 12 400 × 100% = 3%

NowtrythefollowingPracticeExercise

PracticeExercise4Percentages(Answers onpage815)

Inproblems1and2,expressthegivennumbersas percentages.

1.0.057

2.0.374

3.Express20%asadecimalnumber

4.Express 11 16 asapercentage

5.Express 5 13 asapercentage,correctto3 decimalplaces

6.Placethefollowinginorderofsize,thesmallestfirst,expressingeachaspercentages,correctto1decimalplace:

(a) 12 21 (b) 9 17 (c) 5 9 (d) 6 11

7.Express65%asafractioninitssimplestform

8.Calculate43.6%of50kg

9.Determine36%of27mv

10.Calculatecorrectto4significantfigures:

(a)18%of2758tonnes

(b)47%of18.42grams

(c)147%of14.1seconds

11.Express:

(a)140kgasapercentageof1t

(b)47sasapercentageof5min

(c)13.4cmasapercentageof2.5m

12.Acomputerisadvertisedontheinternetat £520,exclusiveofVAT.IfVATispayableat 20%,whatisthetotalcostofthecomputer?

13.Express325mmasapercentageof867mm, correctto2decimalplaces.

14.Whensigninganewcontract,aPremiership footballer’spayincreasesfrom£15,500to £21,500perweek.Calculatethepercentage payincrease,correctto3significantfigures.

15.Ametalrod1.80mlongisheatedanditslength expandsby48.6mm.Calculatethepercentage increaseinlength.

1.4Ratioandproportion

Ratios

Ratioisawayofcomparingamountsofsomething;it showshowmuchbiggeronethingisthantheother. Ratiosaregenerallyshownasnumbersseparatedbya colon(:)sotheratioof2and7iswrittenas2:7and wereaditasaratioof‘twotoseven’.

Herearesomeworkedexamplestohelpusunderstand moreaboutratios.

Problem25. Inaclass,theratiooffemaleto malestudentsis6:27.Reducetheratiotoits simplestform.

Both6and27canbedividedby3

Thus,6:27isthesameas 2:9 6:27and2:9arecalled equivalentratios

Itisnormaltoexpressratiosintheirlowest,orsimplest, form.Inthisexample,thesimplestformis 2:9 which meansforevery2femalesintheclassthereare9male students.

Problem26. Agearwheelhaving128teethisin meshwitha48toothgear.Whatisthegearratio?

Gearratio = 128:48

Aratiocanbesimplifiedbyfindingcommonfactors. 128and48canbothbedividedby2,i.e.128:48isthe sameas64:24

64and24canbothbedividedby8,i.e.64:24isthe sameas8:3

Thereisnonumberthatdividescompletelyintoboth8 and3so8:3isthesimplestratio,i.e. thegearratiois 8:3

128:48isequivalentto64:24whichisequivalentto8:3 8:3isthesimplestform.

Problem27. Awoodenpoleis2.08mlong. Divideitintheratioof7to19.

Sincetheratiois7:19,thetotalnumberofpartsis 7+19=26parts

26partscorrespondsto2.08m = 208cm,hence,1part correspondsto 208 26 = 8

Thus,7partscorrespondsto7 × 8 = 56cm, and19partscorrespondsto19 × 8 = 152cm

Hence, 2.08mdividesintheratioof7:19as56cm to152cm

(Check:56+152mustaddupto208,otherwisean errorwouldhavebeenmade.)

Problem28. Express45pasaratioof£7.65in itssimplestform.

Changingbothquantitiestothesameunits,i.e.topence, givesaratioof45:765

Dividingbothquantitiesby5gives:45:765 ≡ 9:153

Dividingbothquantitiesby3gives:9:153 ≡ 3:51

Dividingbothquantitiesby3againgives:3:51 ≡ 1:17

Thus, 45pasaratioof£7.65is1:17

45:765,9:153,3:51and1:17are equivalentratios and 1:17isthesimplestratio

Problem29. Aglasscontains30mlofwhisky whichis40%alcohol.If45mlofwaterisadded andthemixturestirred,whatisnowthealcohol content?

The30mlofwhiskycontains40%alcohol = 40 100 × 30 = 12ml

After45mlofwaterisaddedwehave30 + 45 = 75ml offluidofwhichalcoholis12ml

Fractionofalcoholpresent = 12 75

Percentageofalcoholpresent = 12 75 × 100% = 16%

NowtrythefollowingPracticeExercise PracticeExercise5Ratios(Answersonpage 815)

1.Inaboxof333paperclips,9aredefective.Expressthenon-defectivepaperclips asaratioofthedefectivepaperclips,inits simplestform.

2.Agearwheelhaving84teethisinmeshwith a24toothgear.Determinethegearratioin itssimplestform.

3.Ametalpipe3.36mlongistobecutinto twointheratio6to15.Calculatethelength ofeachpiece.

4.Inawill,£6440istobedividedbetweenthree beneficiariesintheratio4:2:1.Calculatethe amounteachreceives.

5.Alocalmaphasascaleof1:22,500.The distancebetweentwomotorwaysis2.7km. Howfararetheyapartonthemap?

6.Express130gasaratioof1.95kg

7.Inalaboratory,acidandwateraremixedin theratio2:5.Howmuchacidisneededto make266mlofthemixture?

8.Aglasscontains30mlofginwhichis 40%alcohol.If18mlofwaterisadded andthemixturestirred,determinethenew percentagealcoholiccontent.

9.Awoodenbeam4mlongweighs84kg. Determinethemassofasimilarbeamthat is60cmlong.

10.AnalloyismadeupofmetalsPandQinthe ratio3.25:1bymass.HowmuchofPhasto beaddedto4.4kgofQtomakethealloy.

Directproportion

Twoquantitiesarein directproportion whenthey increaseordecreaseinthe sameratio. Herearesomeworkedexamplestohelpusunderstand moreaboutdirectproportion.

Problem30. 3energysavinglightbulbscost £7.80.Determinethecostof7suchlightbulbs.

12

ElectricalCircuitTheoryandTechnology

If3lightbulbscost£7.80

then1lightbulbcost 7 80 3 =£2.60

Hence, 7lightbulbscost 7 × £2 60 = £ 18 20

Problem31. If56litresofpetrolcosts£59.92, calculatethecostof32litres.

If56litresofpetrolcosts£59.92

then1litreofpetrolcosts 59 92 56 =£1.07

Hence, 32litrescost 32 × 1.07 = £ 34.24

Problem32. Hooke’slawstatesthatstress, σ ,is directlyproportionaltostrain, ε ,withintheelastic limitofamaterial.When,formildsteel,thestress is63MPa,thestrainis0.0003.Determine

(a)thevalueofstrainwhenthestressis42MPa

(b)thevalueofstresswhenthestrainis0.00072

(a)Stressisdirectlyproportionaltostrain. Whenthestressis63MPa,thestrainis0.0003, henceastressof1MPacorrespondstoastrain of 0.0003

63 and thevalueofstrainwhenthestressis

42MPa = 0 0003 63 × 42 = 0.0002

(b)Ifwhenthestrainis0.0003,thestressis63MPa, thenastrainof0.0001correspondsto 63 3 MPa and thevalueofstresswhenthestrainis

0.00072 = 63 3 × 7 2 = 151 2MPa

Problem33. Ohm’slawstatethatthecurrent flowinginafixedresistanceisdirectlyproportional totheappliedvoltage.When90mVisapplied acrossaresistorthecurrentflowingis3A. Determine

(a)thecurrentwhenthevoltageis60mV

(b)thevoltagewhenthecurrentis4.2A

(a)Currentisdirectlyproportionaltothevoltage. Whenvoltageis90mV,thecurrentis3A, henceavoltageof1mVcorrespondstoacurrent of 3 90 A and whenthevoltageis60mV, thecurrent = 60 × 3 90 = 2A

(b)Voltageisdirectlyproportionaltothecurrent. Whencurrentis3A,thevoltageis90mV, henceacurrentof1Acorrespondstoavoltageof 90 3 mV=30mV and whenthecurrentis4.2A, thevoltage = 30 × 4.2 = 126mV

NowtrythefollowingPracticeExercise

PracticeExercise6Directproportion (Answersonpage815)

1.3enginepartscost£208.50.Calculatethecost of8suchparts.

2.If9litresofglosswhitepaintcosts£24.75, calculatethecostof24litresofthesame paint.

3.Thetotalmassof120householdbricksis57.6 kg.Determinethemassof550suchbricks.

4.Hooke’slawstatesthatstressisdirectlyproportionaltostrainwithintheelasticlimitof amaterial.When,forcopper,thestressis60 MPa,thestrainis0.000625.Determine(a)the strainwhenthestressis24MPa,and(b)the stresswhenthestrainis0.0005

5.Charles’slawstatesthatvolumeisdirectly proportionaltothermodynamictemperature foragivenmassofgasatconstantpressure.A gasoccupiesavolumeof4.8litresat330K. Determine(a)thetemperaturewhenthevolumeis6.4litres,and(b)thevolumewhenthe temperatureis396K.

6.Ohm’slawstatesthatcurrentisproportional top.d.inanelectricalcircuit.Whenap.d.of 60mVisappliedacrossacircuitacurrentof 24 μAflows.Determine: (a)thecurrentflowingwhenthep.d.is5V, and (b)thep.d.whenthecurrentis10mA

7.If2.2lb = 1kg,and1lb = 16oz,determinethe numberofpoundsandouncesin38kg(correct tothenearestounce).

8.If1litre = 1.76pints,and8pints=1gallon,determine(a)thenumberoflitresin35 gallons,and(b)thenumberofgallonsin75 litres.

Inverseproportion

Twovariables,xandy,areininverseproportiontoone anotherifyisproportionalto 1 x ,i.e.y α 1 x ory = k x or k = xywherekisaconstant,calledthe coefficientof proportionality

Inverseproportionmeansthatasthevalueofonevariableincreases,thevalueofanotherdecreases,andthat theirproductisalwaysthesame.

Herearesomeworkedexamplesoninverseproportion.

Problem34. Itisestimatedthatateamoffour designerswouldtakeayeartodevelopan engineeringprocess.Howlongwouldthree designerstake?

If4designerstake1year,then1designerwouldtake4 yearstodeveloptheprocess.

Hence,3designerswouldtake 4 3 years,i.e. 1year4 months

Problem35. Ateamoffivepeoplecandeliver leafletstoeveryhouseinaparticularareainfour hours.Howlongwillittakeateamofthreepeople?

If5peopletake4hourstodelivertheleaflets,then1 personwouldtake5 × 4 = 20hours

Hence,3peoplewouldtake 20 3 hours,i.e.6 2 3 hours,i.e. 6hours40minutes

Problem36. TheelectricalresistanceRofa pieceofwireisinverselyproportionaltothe cross-sectionalareaA.WhenA = 5mm 2 , R = 7 2ohms.Determine

(a)thecoefficientofproportionalityand (b)thecross-sectionalareawhentheresistanceis 4ohms.

(a)R α 1 A i.e.R = k A ork = RA.Hence, whenR = 7.2andA = 5,the coefficientofproportionality,k = (7 2)(5) = 36

(b)Sincek = RAthenA = k R WhenR = 4,the crosssectionalarea, A = 36 4 = 9mm2

Problem37. Boyle’slawstatesthatatconstant temperature,thevolumeVofafixedmassofgasis inverselyproportionaltoitsabsolutepressurep.Ifa gasoccupiesavolumeof0.08m 3 atapressureof 1.5 × 10 6 pascals,determine(a)thecoefficientof proportionalityand(b)thevolumeifthepressureis changedto4 × 10 6 pascals.

(a)Vα 1 p i.e.V = k p ork = pV

Hence, thecoefficientofproportionality, k = (1.5 × 10 6 )(0.08) = 0.12 × 10 6

(b) Volume, V = k p = 0.12 × 10 6 4 × 106 = 0.03m3

NowtrythefollowingPracticeExercise

PracticeExercise7Furtherinverse proportion(Answersonpage816)

1.A10kgbagofpotatoeslastsforaweekwitha familyof7people.Assumingalleatthesame amount,howlongwillthepotatoeslastifthere wereonlytwointhefamily?

2.If8mentake5daystobuildawall,howlong wouldittake2men?

3.Ifyisinverselyproportionaltoxandy = 15 3 whenx = 0 6,determine(a)thecoefficientof proportionality,(b)thevalueofywhenxis 1.5,and(c)thevalueofxwhenyis27.2

4.Acartravellingat50km/hmakesajourneyin 70minutes.Howlongwillthejourneytakeat 70km/h?

5.Boyle’slawstatesthatforagasatconstant temperature,thevolumeofafixedmassofgas isinverselyproportionaltoitsabsolutepressure.Ifagasoccupiesavolumeof1.5m 3 atapressureof200 × 10 3 Pascal’s,determine (a)theconstantofproportionality,(b)the volumewhenthepressureis800 × 10 3 Pascalsand(c)thepressurewhenthevolume is1.25m 3 1.5Lawsofindices

14

ElectricalCircuitTheoryandTechnology

Law1:Whenmultiplyingtwoormorenumbers havingthesamebase,theindicesareadded.

Forexample,2 2 × 23 = 22+3 = 25 and54 × 52 × 53 = 54+2+3 = 59

Moregenerally, a m × an = am +n

Forexample,a 3 × a 4 = a 3+4 = a 7

Law2:Whendividingtwonumbershavingthesame base,theindexinthedenominatorissubtractedfrom theindexinthenumerator.

Forexample, 25 23 = 25 3 = 22 and 78 75 = 78 5 = 73

Moregenerally, am an = am n

Forexample, c5 c2 = c5 2 = c3

Law3:Whenanumberwhichisraisedtoapoweris raisedtoafurtherpower,theindicesaremultiplied.

Forexample, 22 3 = 22×3 = 26 and 34 2 = 34×2 = 38

Moregenerally, (a m )n = amn

Forexample, d 2 5 = d 2×5 = d 10

Law4:Whenanumberhasanindexof0,itsvalue is1.

Forexample,3 0 = 1and17 0 = 1

Moregenerally, a 0 = 1

Law5:Anumberraisedtoanegativepoweristhe reciprocalofthatnumberraisedtoapositivepower.

Forexample,3 4 = 1 34 and 1 2 3 = 23

Moregenerally, a n = 1 an Forexample,a 2 = 1 a 2

Law6:Whenanumberisraisedtoafractional powerthedenominatorofthefractionistheroot ofthenumberandthenumeratoristhepower.

Forexample,8 2 3 = 3 √82 = (2)2 = 4 and25 1 2 = 2 √251 = √251 =±5(Notethat √ ≡ 2 √)

Moregenerally, a m n = n √am Forexample,x 4 3 = 3 √x4

Problem38. Evaluateinindexform5 3 × 5 × 52 53 × 5 × 52 = 53 × 51 × 52 (Notethat5means5 1 ) = 53+1+2 = 56 fromlaw1

Problem39. Evaluate 35 34

Fromlaw2: 35 34 = 35 4 = 31 = 3

Problem40. Evaluate 24 24 24 24 = 24 4 fromlaw2 = 20 = 1 fromlaw4 Anynumberraisedtothepowerofzeroequals1

Problem41. Evaluate 3 × 32 34 3 × 32 34 = 31 × 32 34 = 31+2 34 = 33 34 = 33 4 = 3 1 fromlaws1and2 = 1 3 fromlaw5

Problem42. Evaluate 103 × 102 108 103 × 102 108 = 103+2 108 = 105 108 fromlaw1 = 105 8 = 10 3 fromlaw2 = 1 10 +3 = 1 1000 fromlaw5

Hence, 103 × 102 108 = 10 3 = 1 1000 = 0 001

Problem43. Simplify:(a) (2 3 )4 (b) (3 2 )5 expressingtheanswersinindexform.

Fromlaw3: (a) (23 )4 = 23×4 = 212 (b) (3 2 )5 = 32×5 = 310

Problem44. Evaluate: (102 )3 104 × 102

Fromlaws1,2,and3: (102 )3 104 × 102 = 10 (2×3) 10 (4+2) = 106 106 = 106 6 = 100 = 1 fromlaw4

Problem45. Evaluate(a)4 1/2 (b)16 3/4 (c)27 2/3 (d)9 1 2

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“It is an heirloom of my house,” said I. “It was given by my father to my mother when he came to woo her.”

The Englishman raised his eyebrows with an aspect of grave interest.

“Was that so, my young companion? Given by your father to your mother—was that really the case? And set with agates, unless my eyes deceive me.”

“Yes, they are agates.”

“The sight of agates puts me in mind of a ring I had of my old friend, the Sophy. I used always to affect it on the middle finger of the right hand, just as you affect your own, my son, until it was coveted by my sainted mother upon a wet Ash Wednesday.”

Still exhibiting the tokens of a lively regard, the Englishman began to fondle the ring as it lay on my finger.

“An honest band of gold, of a very chaste device. It looks uncommonly choice on the hand of a gentleman. Does it not fit somewhat loosely, my young companion?”

Speaking thus, and before I could suspect his intention, Sir Richard Pendragon drew the ring off my finger. He held it up to the light, and proceeded to examine it with the nicest particularity.

“I observe it was made in Milan,” said he. “It must have lain for years in a nobleman’s family My own was fashioned in Baghdat, but I would say this is almost as choice as the gift of the Sophy. And as I say, my son, it certainly makes an uncommonly fine appearance on the hand of a gentleman.”

Thereupon Sir Richard Pendragon pressed the slender band of metal upon the large fat middle finger of his right hand.

“It comes on by no means so easy as the Sophy’s gift,” said he; “but then, to be sure, my old gossip had a true circumference taken by the court jeweller. I often think of that court jeweller, such an odd, brisk little fellow as ’a was. ’A had a cast in the right eye, and I remember that when he walked one leg went shorter than its

neighbour But for all that ’a knew what an agate was, and his face was as open as a fine evening in June.”

With an air of pleasantry that was impossible to resist, Sir Richard passed his cup and exhorted me to drain it. I drank a little of the wine, yet with some uneasiness, for it was sore to me that my father’s talisman was upon the hand of a stranger.

“I shall thank you, sir, to restore the ring to my care.”

“With all the pleasure in life, my son.” The Englishman took hold of his finger and gave it a mighty pull, but the ring did not yield.

He shook his head and began to whistle dolefully.

“Why, as I am a good Christian man this plaguy ring sticks to my hand like a sick kitten to a warm hearthstone. Try it, my son, I pray you.”

I also took a pull at the ring, which was wedged so firmly upon his hand, but it would not budge.

“This is indeed a terrible matter,” said Sir Richard Pendragon. “What is to be done, young Spaniard?”

He called the innkeeper and bade him bring a bowl of cold water. Into this he dipped his finger; and although he held it in the water for quite a long time, the ring and his right hand could not be induced to part company.

“What is the price you set upon this ring, my young companion?”

“The ring is beyond price—it was my mother’s—and has ever been in the keeping of an ancient house.”

“If it is beyond price there is an end to the question. I was about to offer you money, but I see you have one of those lofty spirits that can brook no vulgar dross. Well, well, pride of birth is a good thing, and money is but little. Yet one who has grown old in the love of virtue would like to requite you in some way. Had we not better throw a main with the dice? If I win I wear the ring for my lawful use; and if I lose you shall have the good tuck that was given to me by the King of Bavaria for helping him against the Dutch.”

I did not accept this suggestion, as you may believe. Yet it gave me sore concern to see my father’s heirloom upon the hand of this foreigner. In what fashion it was to be lured from his finger I was at a loss to know; and in my inexperience of the world I did not know what course to embrace.

CHAPTER V

I HEAR OF THE PRINCESS

U his own part Sir Richard Pendragon showed a wonderful calmness. He wore the ring upon his finger with so great an air, and withal was so polite that the forcing of a quarrel was put out of the question. None the less it was clear that if ever I was to recover my father’s gift it must be at the point of the sword.

It is always claimed, however, by the natives of my province, which in the things of the mind is allowed to be the first in all Spain, that a cool judgment must ride before violence. Therefore I was in no haste to push the matter to an extremity. My mind was set that I could only regain possession of the trinket by an appeal to the sword; that soon or late we must submit ourselves to that arbitrament, but as the night was yet in its youth, I felt there was no need to force the brawl before its season. Thus, nursing my injury in secret I marked the man narrowly as he sat his stool, with his hungry eyes forever trained upon me sideways, and forever glancing down with furtive laughter, while his great lean limbs in their patched, parti-coloured hose, in which the weather had wrought various hues, were sprawled out towards the warmth of the chimney.

As thus he lay it was hard to decide whether he was indeed a king’s son or no more than a fluent-spoken adventurer. And in spite of the flattering opinions he put forward of his own character, I was fain to come to it that the latter conclusion was at least very near to the truth. For one thing, the lack of seriousness in his demeanour did not consort very well with the descendant of princes, whom all the world knows to be grave men. He never so much as looked towards me without a secret light of mirth in his eye; and this I was unable to account for, as for myself I had never felt so grave.

“Sir Richard Pendragon, knight,” said he, for no particular reason, unless it were the love of hearing his own discourse; “of all names I believe that to be the most delectable; for it is the name of a true man, of one addicted to contemplation, and of one who has grown old in the love of virtue. Sir Richard Pendragon, knight—a name is a small thing, but it has its natural music; Sir Richard Pendragon, knight—yes, it runs off the tongue to a tune. I think, my young companion, you have already admired it?”

“Indeed, sir, I have,” said I, with a certain measure of mockery, of which, upon occasion, those of my province are said to be adept in the use. “I conceive it to be a most wonderful name. Have you not said so yourself?”

“If I have, I have,” said he, patting my shoulder with a familiarity for which I did not thank him. “After all, the murder was obliged to come out. Is it the part of valour to shun the truth? My young companion, I feel sure you are one of those who respect that pious opinion that is shared by P. Ovidius Naso and other learned commentators upon the subject. Indeed, it is very well that a name which stands so high in middle Europe is come into this outer part. Quite recently I feared it to be otherwise. I met an itinerant priest, not a month ago, bald, obese, and biblical, who said that to his mind my name was deficient. ‘Fair sir, for what is it celebrated?’ was his question. ‘For what is it celebrated, reverend one?’ was my rejoinder. ‘Why, where can you have lived these virtuous years of yours? It is the name of a notorious pea-nut and straw-sucker.’ ‘That is verily a singular accomplishment,’ said the reverend father in God. ‘Yes, your reverence,’ I answered, ‘this old honest fellow can draw a nut through a straw with the same complacency as a good churchman can draw sack through the neck of a bottle.’ ‘That is indeed remarkable,’ said the reverend father, and proceeded to demonstrate that as pea-nuts were wide and straws were narrow, it was no light matter. ‘Yes, my father,’ said I, ‘that is a very just observation. But I am sure you would be the last to believe that one who has a king’s blood flowing under his doublet would bring his mind to anything trivial.’ ‘Doubtless your view is the correct one,’ said the reverend sceptic, ‘but all the same, I fail to see how a king’s blood would be

able to compass a feat of that nature.’ ‘There is none shall say what a king’s blood will compass,’ was my final rejoinder, ‘for there is a particular genius in it.’ Yet, my young son of the Spains, I have little doubt that the worthy Dominican is still breaking his mind upon this problem behind the walls of Mother Church; and such is the subtlety of these scholars with their thumb rules and their logicality, that presently you shall find that this innocent pleasantry has unhinged the brains of half the clerks in Salamanca.”

“You have indeed a ready wit and a subtle contrivance, sir,” said I at the conclusion of this ridiculous tale, for it was plain that he looked for some such comment upon it.

“You must blame my nation for that. Every Englishman is witty when he has taken wine; he is an especially bright dog in everything after the drinking of beer. You dull rogues of the continent can form no conception of an Englishman’s humour.”

“How comes it, sir, that you find yourself an exile from this land which, by your account of it, is fair unspeakably?”

“It is a matter of fortune,” he made answer.

“Is that to say you are on a quest of fortune?” said I, breathing high at this magic word.

“You have come to the truth,” said he with a sigh and a smile and a sidelong look at the sword that hung by his leg.

“Why then, sir,” I cried with an eagerness I could not restrain, “we are as brothers in this matter. I also am on a quest of fortune.”

My words seemed to jump with the humour of Sir Richard Pendragon. He looked at me long and curiously, with that side glance which I did not find altogether agreeable, stroked his beard as if sunk in deep thought, and said with the gravest air I had heard him use,

“Oh, indeed, my son, is that the case! So you are on a quest of fortune, are you, my son? Well, she is a nice, a proper, and a valiant word.”

“My father was ever the first to allow it,” said I. “She used him ill; his right hand was struck off in a battle at a tender age, but I never heard him complain about her.”

“She hath ever been haughty and distant with old English Dick,” said my companion, sighing heavily; “but you will never hear that true mettle abuse the proud jade. Fortune,” he repeated and I saw his great hungry eyes begin to kindle until they shone like rubies—“oh, what a name is that! She is sweeter in the ears of us of England than is the nightingale. What have we not adventured in thy name, thou perfect one! Here is this Dick, this old red bully, with his dry throat and his sharp ears and his readily watering eye, what hath he not dared for thee, thou dear ungracious one! He has borne his point in every land, from the wall of China to the high Caucasian mountains; from the blessed isles of Britain to farthest Arabia. Who was it drove the Turk out of Vienna with a six-foot pole? Who was it beat the Preux Chevalier off his ground with a short sword? Who was it slew the sultan of the Moriscoes with his own incomparable hand? Who was it, and wherefore was it, my son?”

In this exaltation of his temper he peered at me with his side glance, as though he would seek an answer to a question to which no answer was necessary.

“Why do I handle,” he proceeded, “the sword, the broadsword, the short sword, the sword and buckler, and above all that exquisite invention of God, the nimble rapier of Ferrara steel, with the nice mastery of an old honest blade, but in thy service, thou sweet baggage with thy moist lip and thy enkindling eye?”

“Ah! Sir Englishman,” cried I, feeling, in spite of his rough brogue, the music of his nature, “I love to hear you speak thus.”

“Thirty years have I been at the trade, good Spaniard, and sooner than change it I would die. One hundred towns have I sacked; ten fortunes have I plundered. But by sack they came, and by sack they did depart. It is wonderful how a great nature has a love of sack. Yet I have but my nose to show for my passion. Do you observe its prominent hue, which by night is so luminous that it flames like a beacon to forewarn the honest mariner? Yet to Fortune will we wet

our beards, good Spaniard, for we of England court her like a maiden with a dimple in her cheek.”

Having concluded this declamation, Sir Richard Pendragon called the landlord in a tone like thunder, bade him bring a cup of sherry for my use, and fill up his own, which was passing empty.

“I will bear the charges, lousy one,” said Sir Richard with great magnificence.

“Oh yes, your worship”—the poor innkeeper was as pale as a corpse —“but there is already such a score against your worship—”

“Score, you knave!” Sir Richard rolled his eyes horribly. “Why, if I were not so gentle as a woman I would cut your throat. Score, you dog! Then have you no true sense of delicacy? Now I would ask you, you undershot ruffian with your bleared eyes and your soft chaps, are gentlemen when they sit honouring their mistresses in their own private tavern, are they to be crossed in their sentiments by the lowest order of man? Produce me two pots of sack this minute, or by this hand I will cut a gash in your neck.”

The unlucky wight had fled ere his guest had got half through this speech, which even in my ears was frightful, with such roars of fury was it given. When he returned with two more cups filled with wine, Sir Richard looked towards me and laid his finger to the side of his nose, as though to suggest that he yielded to no man in the handling of an innkeeper.

By the time he had drunk this excellent liquor there came a sensible change in the Englishman’s mien. The poetry of his mood, which had led him to speak of Fortune in terms to kindle the soul, yielded to one more fit for common affairs.

“Having lain in my castle,” said he, “and being well nourished with sack, to-morrow I start on my travels again. Upon pressure I would not mind taking a young squire.”

He favoured me with a look of a very searching character. “I say,” he repeated solemnly, stretching out his enormous legs, “I am minded to take a young squire.”

“In what, sir, would his duties consist?”

“They would be mild, good Don. Assuming that this young squire—if he were a man of birth so much the better—paid me a hundred crowns a year, cleaned my horse of a morning and conversed with me pleasantly in the afternoon, I would undertake to teach him the world.”

“Why, sir,” said I, “surely it would be more fitting if your squire received one hundred crowns from you annually, which might stand as his emolument.”

“Emolument!” said the Englishman, stroking his beard. “One hundred crowns! These be very quaint ideas.”

“Why, sir,” said I, with something of that perspicacity for which our province is famous, “would not your squire have duties to perform, and would they not be worthy of remuneration?”

“Duties!—remuneration!” said the Englishman, stroking his beard furiously “Why, can you not know, good Don, I am in the habit of receiving a thousand guilders per annum for teaching the world to sons of the nobility?”

“Indeed, sir! can a knowledge of the world be of so much worth?”

The Englishman roared at that which he took for my simplicity.

“By my soul!” he exclaimed, “a knowledge of the world is a most desperate science. I have met many learned men in my travels, but that science always beat them. Cæsar was a learned man, but he would have had fewer holes in his doublet had he gone to school earlier. It is a deep science, my son; it is the deepest science of all. What do you know of deceit, my son, you who have never left your native mountains before this morning? You, with the dust of your rustic province upon your boots, what do you know of those who hold you in fair speaking that they may know the better where to put the knife?”

“I confess, sir, I have thought but little of these things,” I said humbly, for my misadventure with the beggar woman was still in my mind, and my mother’s ring was no longer in the keeping of her only son.

“Then you will do well to think upon it, my young companion,” said the Englishman, regarding me with his great red eyes. “You talk of fortune, Spaniard, you who have yet to move ten leagues into the world! Why, this is harebrained madness. You who have not even heard of the famous city of London and the great English nation, might easily fall in with a robber, or be most damnably cheated in a civil affair. Why, you who say ‘if you please’ to an innkeeper might easily lose your purse.”

“I may be ill found in knowledge, sir, but I hope my sword is worthy,” said I, determined that none should contemn my valour, even if my poor mind was to be sneered at.

“Oh, so you hope your sword is worthy, do you now?” The Englishman chuckled furiously as if moved by a conceit. “Well, Master No-Beard, that is a good accomplishment to carry, and I suspect that you may find it so one of these nights when there is no moon.”

All the same Sir Richard Pendragon continued to laugh in his dry manner, and fell again to looking at me sideways. For my life I could not see where was the occasion for so much levity.

“My father has taught me the use of the sword,” said I.

“Oh, so your father has taught you the use of the sword! Well, to judge by the length of your beard, good Don, I am inclined to suspect that your father had a worthy pupil.”

“I hope I may say so.”

“Oh, so you hope you may say so, my son! Well, now, I think you may take it, good Don, from one who has grown old in the love of virtue, that your father would know as much of the sword as a burgomaster knows of phlebotomy. You see, having had his right hand struck off in battle at a tender age, unless he happened to be a most infernally dexterous fellow he forfeited his only means of becoming a learned practitioner.”

The Englishman laughed in his belly.

“My father had excellent precept,” said I, “although, as you say, the Hand of God curtailed his practice.”

“Well now, my son,” said Sir Richard Pendragon, assuming a grave air, which yet did not appear a very sincere one, “he who speaks you is one whose practice the Hand of God has not curtailed. He was proficient with sword and basket in his tenderest infancy. He has played with all the first masters in Europe; he has made it a life study. With all the true principles of this inimitable art he is familiar. He has been complimented upon his talent and genius, natural and acquired, by those whom modesty forbids him to name. And all these stores, my worthy Don, of experience, ensample, and good wit are at your command for the ridiculous sum of an hundred crowns.”

“I have not an hundred crowns in the world, sir,” I confessed with reluctancy, for his arguments were masterful.

“By cock!” he snarled, “that is just as I suspected.”

There could be no mistaking the change in his demeanour when I made this unhappy confession. It caused him to resolve his gross and rough features into some form of contemplation. At last he said, with an eye that was like a weasel’s,—

“What is the sum in your poke, good Spaniard?”

“I have but eight crowns.”

“Eight crowns! Why, to hear your conversation one would think you owned a province.”

“A good sword, a devout heart, and the precepts of my noble father must serve, sir, as my kingdom,” said I, hurt not a little at the remarkable change that had come over him.

“I myself,” said he, “have always been governor and viceregent of that kingdom, and had it not been for a love of canaries in my youth, which in my middle years has yielded to a love of sherris, I must have administered it well. But there is also this essential divergence in our conditions, my son. I am one of bone and sinew, an Englishman, therefore one of Nature’s first works; whereas you, good Don, saving your worshipful presence, are but a mincing and

turgid fellow, as thick in the brains as a heifer, and as yellow in the complexion as a toad under his belly. Your mind has been so depressed by provincial ideas, and your stature so wizened by the sun, that to a liberal purview they seem nowise superior to a maggot in a fig, or a blue-bottle fly in the window of a village alehouse.”

“Sir Englishman,” said I haughtily, for since I had told him I had but eight crowns in the world his manner of speaking had grown intolerable, “I do not doubt that among your own nation you are a person of merit, but it would not come amiss if you understood that you pay your addresses to a hidalgo of Spain. And I must crave leave to assure you that in his eyes one of your nation is but little superior to a heathen Arab who is as black as a coal. At least, I have always understood my father, God keep him! to say this.”

“By my faith, then,” said the Englishman, “even for a Spaniard your father must have been very ill informed.”

“Sir Richard Pendragon,” said I sternly, “I would have you be wary of the manner in which you mention my father.”

“I pray you, brother, do not make me laugh.” He trained his sidelong look upon me. “I have such an immoderately nimble humour—it has ever been the curse of my family from mother to daughter, from father to son—as doth cause the blood to commit all manner of outrages upon mine old head veins. All my ancestors died of a fluxion that did not die of steel. But I tell you, Spaniard, it is as plain as my hand that your father must have been a half-witted fellow to beget such a poor son.”

“Sir Richard Pendragon,” I cried, incensed beyond endurance, “if you abuse my father I will run you through the heart!”

“Well,” said he, “this is good speaking on eight crowns, a provincial accent, and a piece of rusty iron which is fitter to toast half a saddle of mutton than to enhance the scabbard of a gentleman. And if you make this speaking good, why, it will be still better. For this is a very high standard, brother, you are setting up, and I doubt me grievously whether even the Preux Chevalier would be able to maintain it.”

He concluded with such an insolent and unexpected roar of laughter as made me grow furious.

“I would have you beware, sir!” I cried. “Were you twice as gross in your stature and three times as rude, I run you through the heart if again you contemn the unsullied name of my noble father.”

“Your father was one-handed,” said this gigantical ruffian, looking at me steadily. “He was as stupid in his wits as a Spanish mule, and I spit in the face of the unbearded child that bears his name.”

CHAPTER VI

OF A PRIVATE BRAWL. I TAKE PROFIT AT THE COST OF REPUTATION

B I could draw my sword my challenger was on his feet, had kicked away the stool on which he had sat, and had bared his own weapon. I was so overcome with fury that I could not stay to mark his enormous stature, yet his head seemed to live among the hams in the roof.

This was the first occasion I had drawn my sword in a quarrel, but I needed to ask no better. The pure reputation of this noble heart I was defending nerved my right arm with unimaginable strength. Besides, I was twenty-one years old, well grown and nourished for one of my nation. My blade was of an ancient pattern, but a true Toledo of the first quality, and many high deeds of the field had been wrought thereby The Englishman towered above me in the extremity of his stature, but had he been of twice that assemblance, in my present mood I would not have feared him. For, as I was fain to believe, some of the hardiest fighting blood of our northern provinces was in my veins. This was my first duello; but you must not forget, reader, that my father had instructed me how to bear my point, how to thrust, how to receive, and, above all, how to conduct the wrist as laid down by the foremost practice.

We spent little time in courtesies, for my anger would not permit them. At once I ran in upon my adversary, thrusting straight at his heart. Yet he received my sword on his own with a skill that was truly wonderful, and turned it aside with ease. All the power I possessed was behind it, yet he cast it off almost as complacently as if he had been brushing away a mosquito. The sting of this failure and his air of disdain caused me to spring at him like a cat, yet, I grieve to say, without its wariness, for, do what I would, I was unable to come near

him. He saved every stroke with a most marvellous blade, not once moving his wrist or changing his posture. After this action had proceeded for some minutes I was compelled to draw off to fetch my breath; whereon said my adversary with a snarl of contempt that hurt me more than my impotence:—

“I wish, my son, you would help me to pass the time of the day.”

My instant response was a most furious slash at his head, although it is proper to mention that this method was not recommended in the rules of the art as expounded by the illustrious Don Ygnacio. But I grieve to confess that rage had overmastered me. Yet Sir Richard Pendragon evaded this blow as dexterously as he had evaded the others.

“Come, brother,” he said; “even for a Spaniard this is futility. This is no more than knife work. I am persuaded your father was a butcher, and owed his entire practice to the loins of the Galician hog.”

Such derision galled me worse than a thrust from his sword. Casting away all discretion I ran in upon him blindly, for at that moment I was minded to make an end one way or another.

“Worse and worse,” said he. “You bear your blade like a clergyman’s daughter. Still, do not despair, my young companion; perhaps you will make better practice for my left hand.”

As he spoke, to my dismay he changed his weapon from his right hand to his left, and parried me with the same contemptuous dexterity. Suddenly he made a strong parade, and in the next instant I felt the point of his sword at my breast.

“Your father must have been a strangely ignorant man, even for a Spaniard,” he said. “I do not wonder that he lost his right hand at an early age. You have as little defence as a notorious cutpurse on his trial. Any time these five minutes you must have been slain.”

Then it was I closed my eyes in the extremity of shame and never expected to open them again. But to my astonishment the forces of nature continued to operate, and soon, in a vertigo of fear and anger, I was fain to look for the cause. It seemed that my enemy had lowered his point and drawn off. Plainly he intended to use me as a

cat uses a mouse, for his private pleasure. For that reason I fell the harder upon him, since I knew my life to be forfeit, and I had an instinct that the more furiously it was yielded the less should I know of a horrid end.

“I will now slit your doublet, my son,” said Sir Richard Pendragon. “Have you a favourite rib you would care to select? What of the fifth?”

Without more ado he began cutting my doublet with a dexterity that was amazing. His point flashed here and there across my breast and seemed to touch it in a thousand places; yet, although the old leather was pierced continually, no hap was suffered by my skin.

“If only I had my lighter and more fanciful blade of Ferrara here,” he said in the midst of a thousand fanfaronades and brandishments, “I would flick every button off your doublet so nicely as a tailor.”

“Kill me!” I cried, flinging myself upon his blade.

I made such a terrific sweep with my sword that it whistled through the air, and was like to cut off his head. Instead, however, of allowing it to do so, he met it with a curious turn of the wrist, and the weapon was hurled from my grasp.

As I stood before him panting and dishevelled, and young in the veins and full in them too, I seemed to care no more than a flake of snow for what was about to occur. I could but feel that I had traduced my father’s reputation, and had cast a grievous slur upon his precept. The blood was darkening my eyes and singing in my ears, but quite strangely I was not minding the blade of my enemy. That which was uppermost in my mind was the landlord’s opinion that he was the Devil in Person.

Upon striking my sword to the ground he bade me remark his method of disarmament.

“It approaches perfection so nearly,” said he, “as aught can that is the offspring of the imperfection of man. It is the fruit of a virtuous maturity; it is the crown of artifice; consider all the rest as nought. For I do tell thee, Spaniard, this piece of espièglerie, as they say at Paris, divides one of God’s own good swordsmen from the vulgar

herd of tuck-pushers or the commonalty And, mark you, it was all done with the left hand.”

While awaiting with as much composure as I could summon that stroke which was to put me out of life, there happened a strange thing. There had come into the room, unobserved by us both, the tap-wench to the inn. And in a moment, seeing what was toward, this brave little creature, not much bigger than a stool, and as handsome and flashing a quean as ever I saw, ran between me and the sword of my adversary.

“Hold, you bloody foreign man!” she cried imperiously

“Nay, hold yourself, you neat imp,” said the Englishman, catching her round the middle by his right arm, and lightly hoisting her a dozen paces as though she had been a sack of feathers. Yet he had made but a poor reckoning if he thought he could thus dispose of this fearless thing. For his wine cup, half full of sherry, which had been set in the chimney-place out of the way of hap, was to her hand. She picked it up, and hurled the pot and its contents full in the face of the giant.

“Take it, you wicked piece of villainy!” she cried.

Now, by a singular mischance the edge of the cup struck Sir Richard Pendragon on the forehead. It caused a wound so deep that his blood was mingled with the excellent wine. Together they flowed into his eyes and down his cheeks, and so profusely that they stained his doublet and dripped upon the floor. And the courageous girl, seeing my enemy’s discomfiture, for what with the liquor and what with his gore he was almost blind for the nonce, she darted across the room and picked up my sword. With a most valiant eagerness she pressed it into my hand.

“Now, young señor gentleman, quick, quick!” she cried. “Have at him and make an end of him!”

“Alack, you good soul,” said I, “this cannot be. I am the lawful prize of my adversary. God go with you, you kind thing.”

I cast the sword to the ground.

“Then oh, young master, you are a very fool.” Tears sprang to the eyes of the honest girl and quenched her fiery glances.

However, so dauntless was the creature in my cause that she picked up my sword again, and crying, “I myself will do it, señor,” actually had at the English barbarian with the greatest imaginable valiancy.

In the meantime the giant had been roaring at his own predicament in the most immoderate fashion. For, on feeling his head, and discovering that the stream that trickled into his eyes was a compound of elements so delectable, he cast forth his tongue at it in a highly whimsical manner, and drew as much into his mouth as he could obtain.

“I have my errors,” he cried, rocking with mirth; “but if a wanton disregard of God’s honest sherris be there among, when he dies may this ruby-coloured one be called to the land of the eternal drought. Jesu! what a body this Pendragon azure gives it. ’Tis choicer than Tokay out of the skull of a Mohammedan. When the hour comes to invest me in my shell, I will get me a tun of sherris and sever a main artery, and I will perish by mine own suction.”

He had scarcely concluded these comments when the brave little maid had at him with my sword. Expecting no such demonstration on the part of one not much taller than his leg, it needed all his adroitness of foot, which for one of his stature was indeed surprising, to save the steel from his ribs. And so set was the creature on making an end of him that the force with which she dashed at his huge form, and yet missed it, carried her completely beyond her balance. With another of his mighty roars, the English giant seized her by the nape with his right hand, and held her up in the air by the scruff, so curiously as if she had been a fierce little cat that had flown at him.

“Why, thou small spitfire,” he said, “thou art even too slight to be cracked under mine heel. Thou pretty devil, I will buss thee.”

“I will bite off the end of your nose, you bloody-minded villain,” cried the little wench, struggling frantically in his gripe.

“Nay, why this enmity, pretty titmouse,” said the giant, “seeing that I have a mind to fondle thee for thy valour?”

“You would slay the young gentleman señor, you wicked cut-throat villain, you!”

“Nay, by my hand I will not, if you will give me twenty honest busses, you neat imp, to heal my contusion.”

“You swear, Englishman, upon your wicked beard, the young señor gentleman shall come to no hurt if I kiss you?”

“I will swear, thou nice hussy, by the bones of all my ancestors in their Cornish cemetery, that young Don Cock-a-hoop shall go uncorrected for all his sauciness and pretension. With eight crowns in his wallet and a most unfathomable ignorance he drew his tuck on a right Pendragon. But so much effrontery shall go unvisited, mark you, at the price of twenty honest busses from those perfect lips of thine. If thou art not the most perfect thing in Spain, I am little better than a swaggerer.”

“Put me down then, Englishman,” said the little wench as boldly as an ambassador; “and do you give the young gentleman señor his sword.”

“So I will; but I would have you remark it, pretty titmouse, that I will be embraced with all the valiancy of thy nature. Ten on each side of my royal chaps, and one for good kindness right i’ th’ middle.”

“Give the young gentleman señor his sword, then, you English villain.”

So had this matter accosted the humour of Sir Richard Pendragon that he obeyed her.

“Take it, young Spaniard,” said he with a magnificent air; “and do you consider it as your first lesson in the affairs of the world. I do perceive two precepts to whose attention your noble father does not appear to have directed you. The first is, never draw upon the premier swordsman of his age, so long as life hath any savour in it; and for the other, never lack the favour of a farthingale. Do I speak sooth, good girl?”

“Yes, you do, you large villain,” said the little creature, with her two fierce eyes as black as sloes. “And now I will kiss you quickly, so that I may have done. I shall scarcely be able to chew so much as a piece of soft cheese for a month after it.”

The Englishman seated himself upon his stool, and set her upon his knees.

“Begin upon the right, my pretty she, slowly, purposefully, and with valiancy. I would as lief have your lips as a bombard of sherris. If it were not for one Betty Tucker, a dainty piece at the ‘Knight in Armour’ public-house hard by to the town of Barnet, in the kingdom of Great Britain, I would bear you at my saddle-bow all the way back to our little England, and marry you at the church of Saint Clement the Dane, which is in London city. For next to sack I love valour, and next to valour I love my soul. Now then, thou nice miniard, I must taste thy lips softly, courteously, but yet with valiancy as becomes thy disposition.”

It was never my fortune to behold a sight more whimsical than that of this monstrous fellow seated with the blood still trickling down to his chin, while this little black-eyed wench, not much bigger than his fist, with her skin the colour of a walnut, her hair hanging loose, and her rough clothes stained and in tatters, dealt out her kisses first to one side of his ugly mouth and then to the other, yet making as she did so lively gestures of disgust.

“Courteously, courteously!” cried the giant. “Let us have no unmannerly haste in this operation, or I will have them all over again.”

“Nay, you shall not; I will take heed of that. That is fifteen. Another ten, you foreign villain, would give me a canker in my front teeth.”

“Nay, that is but fourteen, my pretty mouse. Here we have the fifteenth. Courteously, courteously, do I not tell thee. See to it that it is so long drawn out that I may count nine.”

“There’s twenty, you large villain!” cried the little creature in huge disgust, and slipping off his knee as quickly as a lizard.

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